Probability Estimation with Truncated Inverse Binomial Sampling

Estimation of a probability in inverse binomial sampling under normalized linear-linear and inverse-linear loss

Sequential estimation of the success probability $p$ in inverse binomial sampling is considered in this paper. For any estimator $\hat p$, its quality is measured by the risk associated with normalized loss functions of linear-linear or inverse-linear form. These functions are possibly asymmetric, with arbitrary slope parameters $a$ and $b$ for $\hat p<p$ and $\hat p>p$ respectively. Interest in these functions is motivated by their significance and potential uses, which are briefly discussed. Estimators are given for which the risk has an asymptotic value as $p$ tends to $0$, and which guaran…

A Theory of Truncated Inverse Sampling

In this paper, we have established a new framework of truncated inverse sampling for estimating mean values of non-negative random variables such as binomial, Poisson, hyper-geometrical, and bounded variables. We have derived explicit formulas and computational methods for designing sampling schemes to ensure prescribed levels of precision and confidence for point estimators. Moreover, we have developed interval estimation methods.

Bayesian Point Estimation and Predictive Density Estimation for the Binomial Distribution with a Restricted Probability Parameter

In this paper, we consider Bayesian point estimation and predictive density estimation in the binomial case. After presenting preliminary results on these problems, we compare the risk functions of the Bayes estimators based on the truncated and untruncated beta priors and obtain dominance conditions when the probability parameter is less than or equal to a known constant. The case where there are both a lower bound restriction and an upper bound restriction is also treated. Then our problems are shown to be related to similar problems in the Poisson case. Finally, numerical studies are presen…

为正、逆贝叶斯问题设计截断先验

Designing truncated priors for direct and inverse Bayesian problems

本文针对带函数型未知量的贝叶斯逆问题，建立了带截断高斯级数先验、满足一般光滑性假设的线性逆问题后验收缩理论，明确了截断点在正、逆问题中的内在作用，推动了贝叶斯逆问题理论发展。

Full adaptation to smoothness using randomly truncated series priors with Gaussian coefficients and inverse gamma scaling

We study random series priors for estimating a functional parameter (f\in L^2[0,1]). We show that with a series prior with random truncation, Gaussian coefficients, and inverse gamma multiplicative scaling, it is possible to achieve posterior contraction at optimal rates and adaptation to arbitrary degrees of smoothness. We present general results that can be combined with existing rate of contraction results for various nonparametric estimation problems. We give concrete examples for signal estimation in white noise and drift estimation for a one-dimensional SDE.

Early stopping for statistical inverse problems via truncated SVD estimation

We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension $D$. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level $\widehat m\in\{1,\ldots,D\}$ only based on the knowledge of the first $\widehat m$ singular values and vectors. We analyse in detail whether sequential {\it early stopping} rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching…

Bayesian inverse problems with partial observations

We study a nonparametric Bayesian approach to linear inverse problems under discrete observations. We use the discrete Fourier transform to convert our model into a truncated Gaussian sequence model, that is closely related to the classical Gaussian sequence model. Upon placing the truncated series prior on the unknown parameter, we show that as the number of observations $n\rightarrow\infty,$ the corresponding posterior distribution contracts around the true parameter at a rate depending on the smoothness of the true parameter and the prior, and the ill-posedness degree of the problem. Correc…

关于无限维贝叶斯逆问题的卡尔文-洛埃夫展开注记

A note on the Karhunen-Loève expansions for infinite-dimensional Bayesian inverse problems

本文探讨了截断卡尔文-洛埃夫展开在无限维逆问题中的应用，证明在特定条件下，误差界可无需解知即可估计。

Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems

We consider the estimation of quadratic functionals in a Gaussian sequence model where the eigenvalues are supposed to be unknown and accessible through noisy observations only. Imposing smoothness assumptions both on the signal and the sequence of eigenvalues, we develop a minimax theory for this problem. We propose a truncated series estimator and show that it attains the optimal rate of convergence if the truncation parameter is chosen appropriately. Consequences for testing problems in inverse problems are equally discussed: in particular, the minimax rates of testing for signal detection …

Parameter Collapse due to the Zeros in the Inverse Condition

Helton, Lasserre, and Putinar (2008, Ann. Probability; arXiv:math/0702314) expose the relationship between three properties of a measure: the conditional triangularity property of the associated orthogonal polynomials, the zeros in the inverse condition of the truncated moment matrix, and conditional independence. The purpose of this article is to provide examples of parameter collapse to product structure given that the zeros in the inverse condition holds up to some degree d. Specifically, start with a parameterized family of probability density functions; require that the zeros in the inver…